Math Problem Statement

solve

What line should you draw on the graph of y = x^2 + 8x - 3 to solve the equation x^2 + 10x - 12 = 0 graphically?

Solution

To solve the equation x2+10x12=0x^2 + 10x - 12 = 0 graphically using the graph of y=x2+8x3y = x^2 + 8x - 3, you need to draw a straight line y=mx+cy = mx + c that represents the difference between the two equations. Here's how to approach this step-by-step:

  1. Set the equations equal to each other: We want to find the points of intersection between y=x2+8x3y = x^2 + 8x - 3 and the line representing x2+10x12x^2 + 10x - 12. Since we already have x2+10x12=0x^2 + 10x - 12 = 0, we can rearrange it: x2+8x3=x2+10x12x^2 + 8x - 3 = x^2 + 10x - 12

  2. Simplify the equation: Subtract x2+8x3x^2 + 8x - 3 from both sides: 0=2x90 = 2x - 9

  3. Express this as a line equation: Rearrange 2x9=02x - 9 = 0 into the slope-intercept form y=mx+cy = mx + c: y=2x9y = 2x - 9

  4. Final Answer: The line you should draw on the graph of y=x2+8x3y = x^2 + 8x - 3 to solve x2+10x12=0x^2 + 10x - 12 = 0 is: y=2x9y = 2x - 9

Would you like me to elaborate further or assist with graphing this solution?

Questions for Deeper Understanding:

  1. What is the significance of the intersection points of the parabola and the line?
  2. How can you verify the solution of x2+10x12=0x^2 + 10x - 12 = 0 graphically after plotting the line?
  3. What happens if the straight line does not intersect the parabola?
  4. How would you determine the slope and yy-intercept of the line algebraically in a different problem?
  5. How does the graphing method compare to solving the quadratic equation algebraically?

Tip:

Graphical solutions are especially useful for visualizing roots of equations and understanding their behavior in relation to other functions!

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Math Problem Analysis

Mathematical Concepts

Graphical Solution of Equations
Linear Equations
Quadratic Equations

Formulas

Linear equation formula y = mx + c
Equating two functions to find intersections

Theorems

Concept of intersections between graphs

Suitable Grade Level

Grades 8-10